AN IN SOLID-STATE ELECTROLYTES

AN AB INITIO INVESTIGATION OF STRUCTURE-FUNCTION RELATIONSHIPS

IN SOLID-STATE ELECTROLYTES

A Thesis

Presented to

the Faculty of the Department of Electrical and Computer Engineering

and the Department of Chemical and Biomolecular Engineering

University of Houston

In Partial Fulfillment

of the Requirements for the Degree

Bachelor of Science

in Electrical Engineering

by Audrey Elyse Wang

May 2019

AN AB INITIO INVESTIGATION OF STRUCTURE-FUNCTION RELATIONSHIPS

IN SOLID-STATE ELECTROLYTES

______________________________

Audrey Elyse Wang

Approved:

Committee Members:

______________________________ Suresh K. Khator, Associate Dean Cullen College of Engineering

______________________________ Yan Yao, Ph.D. Chair of the Committee Department of Electrical and Computer Engineering

______________________________ Lars Grabow, Ph.D. Department of Chemical and Biomolecular Engineering

______________________________ Fritz Claydon, Ph.D. Department of Electrical and Computer Engineering The Honors College ______________________________ Badri Roysam, Ph.D. and Chair Department of Electrical and Computer Engineering

iv

Acknowledgments

First and foremost, I would like to thank Professor Yan Yao for giving me the opportunity

to venture out of electrical engineering and into the realm of materials science and

electrochemistry. Through this experience studying battery materials, I have discovered that

energy storage is a passion of mine that I will continue to pursue throughout my career.

I would also like to thank Professor Lars Grabow and the rest of the UH Computational

Catalysis and Interface Chemistry Group for welcoming me into the group and for their

invaluable expertise regarding density functional theory. It was my absolute pleasure to work

with such joyful and supportive colleagues and friends.

I am also thankful for Professor Fritz Claydon and the mentorship he has provided to me

since my first year at the University of Houston. His belief in me as a freshman helped me to

build confidence in my own abilities and to accomplish far more in my undergraduate career

than I ever imagined.

My sincere thanks also go out to Karun Kumar Rao, without whom my completion of this

research project would have been impossible. Karun coached me and challenged me, and for

his guidance, I am deeply grateful.

Finally, I want to thank my parents for fostering my dream of becoming an engineer since

I was a child and my fiancé for always being available to be my sounding board and for being

an unwavering source of strength and motivation.

v

An Ab Initio Investigation of Structure-Function Relationships in Solid-State Electrolytes

An Abstract

of a

Thesis

Presented to

the Faculty of the Department of Electrical and Computer Engineering

and the Department of Chemical and Biomolecular Engineering

University of Houston

In Partial Fulfillment

of the Requirements for the Degree

Bachelor of Science

in Electrical Engineering

by Audrey Elyse Wang

May 2019

vi

Abstract

Solid-state electrolytes (SSEs), or superionic conductors, are a promising method of energy

storage and a safer alternative to conventional Li-ion batteries. However, the ionic

conductivities of most known SSEs, a characteristic integral to battery performance, are not

yet commercially competitive. Ionic conductivity in SSEs is often achieved through the

interstitial hopping of the mobile cation, so understanding the energetics of the crystal

structure is important. The objective of this thesis is to use density function theory (DFT) to

investigate the relationships between crystal structure and ionic conductivity of SSEs.

Activation energies were calculated using DFT and nudged elastic band theory for sulfide and

oxide frameworks with either lithium or sodium cations. The energy pathways generated in

this study were consistent with previous findings that materials with BCC structures have the

lowest energy barriers and thus have the highest ionic conductivities due to their homogenous

tetrahedral sites.

vii

Table of Contents

Acknowledgments ............................................................................................................................... iv

Abstract ................................................................................................................................................. vi

Table of Contents ............................................................................................................................... vii

List of Figures ...................................................................................................................................... ix

Chapter One – Introduction ............................................................................................................... 1

1.1 Motivation for Studying Battery Energy Storage Systems ................................................... 1

1.2 Ion Transport in Batteries......................................................................................................... 4

1.3 Using Density Functional Theory to Investigate Ionic Conductivity ................................ 6

1.4 Project Statement & Scope ....................................................................................................... 9

Chapter Two – Methods ................................................................................................................... 10

2.1 Selecting Crystal Structures ..................................................................................................... 10

2.2 Interstitial Sites and Ion Hopping Paths............................................................................... 11

2.3 DFT Workflow ......................................................................................................................... 13

2.4 Nudged Elastic Band Calculations ........................................................................................ 15

Chapter Three – Results & Discussion ........................................................................................... 16

3.1 Lithium Cations in a Sulfide Framework (Li-S) ................................................................... 16

3.1.1 Li-S Body-Centered Cubic .............................................................................................. 16

3.1.2 Li-S Face Centered Cubic ................................................................................................ 17

3.1.3 Li-S Hexagonal Close Packing ........................................................................................ 18

3.1.4 Activation Energy for Li-ion Migration in S Anion Framework vs. Volume ......... 21

3.2 Sodium Cations in a Sulfide Framework (Na-S) ................................................................. 23

viii

3.2.1 Na-S Body-Centered Cubic ............................................................................................ 23

3.2.2 Na-S Face-Centered Cubic .............................................................................................. 24

3.2.3 Na-S Hexagonal Close Packing ...................................................................................... 25

3.2.4 Activation Energy for Na-ion Migration in S Anion Framework vs. Volume ........ 26

3.3 Lithium Cations in an Oxide Framework (Li-O) ................................................................ 27

3.3.1 Li-O Body-Centered Cubic ............................................................................................. 27

3.3.2 Li-O Face-Centered Cubic .............................................................................................. 28

3.3.3 Li-O Hexagonal Close Packing ...................................................................................... 28

3.3.4 Activation Energy for Li-ion Migration in O Anion Framework vs. Volume ........ 30

3.4 Sodium Cations in an Oxide Framework (Na-O) ............................................................... 31

3.4.1 Na-O Body-Centered Cubic ........................................................................................... 31

3.4.2 Na-O Face-Centered Cubic ............................................................................................ 32

3.4.4 Activation Energy for Na-ion Migration in O Anion Framework vs. Volume ...... 34

3.5 Limitations of Current Results ............................................................................................... 34

Chapter Four – Summary and Conclusions ................................................................................... 35

References ........................................................................................................................................... 38

Appendix ............................................................................................................................................. 41

ix

List of Figures

Figure 1. High-level overview of how ions and electrons move in lithium-ion batteries. ......... 2

Figure 2. Tetrahedral and an octahedral interstitial site. The tetrahedral site (a) has four anions

around a cation, and the octahedral site (b) has six anions around a cation. Anions in blue,

cations in gold. ............................................................................................................................... 4

Figure 3. High-level overview of the algorithm used for DFT energy relaxations. Note that

𝐹𝑖 − 𝐹𝑖 − 1 < 𝐹𝑐𝑢𝑡𝑜𝑓𝑓 means the forces of the previous iteration (𝑖 − 1) subtracted

from the current iteration 𝑖 must be below a specified cutoff value. .................................... 7

Figure 4. Crystal structure models and energy landscape plots from “Design principles for

solid-state lithium superionic conductors” by Wang et al. for BCC, FCC, and HCP crystal

structures composed of a sulfur anion lattice (yellow) and Li-ion charge carriers (green)

[6]. .................................................................................................................................................... 8

Figure 5. Anion sublattice mapping. a) Existing superionic conductor Li10GeP2S12 (LGPS) [12].

b) LGPS with Ge (blue) and P (green) removed. c) LGPS ultimately maps closely to a

single bcc unit cell of sulfur anions with one lithium ion in a tetrahedral site [13]. Sulfur

anions in yellow, Li-ions in purple. ............................................................................................. 9

Figure 6. Li-ion migration between tetrahedral sites T1 and T2 in a primitive BCC sulfur cell.

........................................................................................................................................................ 11

Figure 7. Li-ion migration from tetrahedral site T1 to octahedral site O1 and finally to

tetrahedral site T2 in a primitive FCC sulfur cell. ................................................................... 12

Figure 8. Li-ion migration in a primitive HCP sulfur cell. There are three paths for ionic

interstitial hopping: tetrahedral to tetrahedral (T1-T3), tetrahedral to octahedral to

tetrahedral (T1-O1-T2), and octahedral to octahedral (O1-O2). ......................................... 12

Figure 9. Illustration of how VASP incorporates a uniform background charge. .................... 14

x

Figure 10. Example of how an Li-ion migration path is interpolated for an NEB calculation.

........................................................................................................................................................ 15

Figure 11. Energy landscapes for a Li-ion migrating through a BCC S anion sublattice via a T-

T path. ........................................................................................................................................... 16

Figure 12. Findings from Wang et al. of the energy landscapes for a Li-ion migrating through

a BCC S anion sublattice via a T-T path [6]. ........................................................................... 16

Figure 13. Energy landscapes for a Li-ion migrating through an FCC S anion sublattice via a

T-O-T path. .................................................................................................................................. 17


an FCC S anion sublattice via a T-O-T path [6]. .................................................................... 18

Figure 15. Energy landscapes for a Li-ion migrating through an HCP S anion sublattice via a

T-T path. ....................................................................................................................................... 18


an HCP S anion sublattice via a T-T path [6]. ......................................................................... 19

Figure 17. Energy landscapes for a Li-ion migrating through an HCP S anion sublattice via an

O-O path. ..................................................................................................................................... 19


an HCP S anion sublattice via an O-O path [6]. ..................................................................... 20

Figure 19. Energy landscapes for a Li-ion migrating through an HCP S anion sublattice via a

T-O-T path. .................................................................................................................................. 20


an HCP S anion sublattice via a T-O-T path [6]. .................................................................... 21

Figure 21. Energy barriers versus cell volume per S anion for various Li-S crystal structures.

The dotted lines show the expected trends. ............................................................................ 22

xi

Figure 22. Findings from previous work of the energy barriers versus cell volume per S anion

for various Li-S crystal structures [6]. ....................................................................................... 23

Figure 23. Energy landscapes for a Na-ion migrating through a BCC S anion sublattice via a

T-T path. ....................................................................................................................................... 24

Figure 24. Energy landscapes for a Na-ion migrating through an FCC S anion sublattice via a

T-O-T path. .................................................................................................................................. 24

Figure 25. Energy landscapes for a Na-ion migrating through an HCP S anion sublattice via a

T-T path. ....................................................................................................................................... 25

Figure 26. Energy landscapes for a Na-ion migrating through an HCP S anion sublattice via

an O-O path. ................................................................................................................................ 26

Figure 27. Energy landscapes for a Na-ion migrating through an HCP S anion sublattice via a

T-O-T path. .................................................................................................................................. 26

Figure 28. Energy barriers versus cell volume per S anion for various Na-S crystal structures.

........................................................................................................................................................ 27

Figure 29. Findings from Wang et al. of the energy barriers versus cell volume per S anion for

various Na-S crystal structures [6]. ............................................................................................ 27

Figure 30. Energy landscapes for a Li-ion migrating through a BCC O anion sublattice via a

T-T path. ....................................................................................................................................... 28

Figure 31. Energy landscapes for a Li-ion migrating through an FCC O anion sublattice via a

T-O-T path. .................................................................................................................................. 28

Figure 32. Energy landscapes for a Li-ion migrating through an HCP O anion sublattice via a

T-T path. ....................................................................................................................................... 29

Figure 33. Energy landscapes for a Li-ion migrating through an HCP O anion sublattice via

an O-O path. ................................................................................................................................ 29

xii

Figure 34. Energy landscapes for a Li-ion migrating through an HCP O anion sublattice via a

T-O-T path. .................................................................................................................................. 30

Figure 35. Energy barriers versus cell volume per O anion for various Li-O crystal structures.

........................................................................................................................................................ 30

Figure 36. Findings from Wang et al. of the energy barriers versus cell volume per O anion for

various Li-O crystal structures [6]. ............................................................................................ 31

Figure 37. Energy landscapes for a Na-ion migrating through a BCC O anion sublattice via a

T-T path. ....................................................................................................................................... 32

Figure 38. Energy landscapes for a Na-ion migrating through an FCC O anion sublattice via a

T-O-T path. .................................................................................................................................. 32

Figure 39. Energy landscapes for a Na-ion migrating through an HCP O anion sublattice via

a T-T path. .................................................................................................................................... 33


an O-O path. ................................................................................................................................ 33


a T-O-T path. ............................................................................................................................... 33

Figure 42. Energy barriers versus cell volume per O anion for various Na-O crystal structures.

........................................................................................................................................................ 34

1

Chapter One – Introduction

1.1 Motivation for Studying Battery Energy Storage Systems

It is well known that anthropogenic global climate change is drastically altering the planet.

As the global population grows, the demand for energy, the excavation of non-renewable

resources, and the production of carbon emissions also rise. Worldwide dependence on fossil

fuels is clearly the primary factor behind climate change, so some of the main challenges of

this century are transitioning global energy dependence to sustainable energy sources and

starkly reducing energy consumption.

Two prominent strategies for increasing sustainability have emerged: first, generating

electricity using renewable energy sources and, second, powering vehicles with sustainably

generated electricity. However, the success of both renewable energy and electric vehicles

(EVs) depends on energy storage development. In the case of renewable energy, the two

frontrunners to replace traditional fossil fuels are wind and solar energy, but the sporadic

nature of wind and sunlight render these sources unable to readily meet consumer needs. In

fact, the rapid integration of renewable energy sources without supplementary storage systems

could be catastrophic; it has been estimated that if renewables grow to constitute 20% or more

of the power supply, then the grid will become destabilized [1]. Similarly, the marketability of

electric vehicles depends entirely on the battery system capacity and cost. Therefore,

developing high-capacity, cost-effective, and environmentally-sound energy storage systems is

one of the most pressing issues in the engineering community.

Although no one energy storage device can fit all possible application requirements, Li-

ion batteries are prime candidates to solve many of the world’s storage needs. Being the most

electropositive element and extremely lightweight, lithium makes for batteries with high

voltages and high volumetric energy densities, which are especially desirable qualities for grid-

2

scale energy storage [2]. For EVs, rapid charge and discharge capabilities are necessary and can

be made possible with Li-ion batteries due to the small radius, and thus fast diffusion, of

lithium ions [2]. However, conventional Li-ion batteries suffer from high flammability as was

seen in 2016 when Samsung Galaxy Note 7 phones exploded as a result of faulty Li-ion

batteries. A failure on the grid- or vehicle-scale would be much more disastrous due to the

large amount of energy being stored, so safety is a primary concern for battery development.

Batteries are energy storage devices that consist of an anode, a cathode, and an electrolyte,

as seen in Figure 1. When a battery is connected to a circuit, a pair of redox reactions occurs

on the anode and cathode. For example, in a lithium-ion battery, lithium deintercalates from

the graphite anode to yield lithium cations and free electrons. Since the electrolyte is

electronically insulating, the electrons travel externally through the circuit, providing electricity

to the device, and the ions travel through the electrolyte, where they are reduced at the cathode.

Figure 1. High-level overview of how ions and electrons move in lithium-ion batteries.

Generally speaking, battery performance is judged based on four main attributes: voltage,

current capacity, energy density, and power density.

3

Voltage (𝑉𝑐𝑒𝑙𝑙) is the electric potential difference between the anode and the cathode, as

shown in equation 1,

𝑉𝑐𝑒𝑙𝑙 = 𝐸𝑐𝑎𝑡ℎ𝑜𝑑𝑒 − 𝐸𝑎𝑛𝑜𝑑𝑒, (1)

where 𝐸𝑐𝑎𝑡ℎ𝑜𝑑𝑒 is the potential of the cathode, and 𝐸𝑎𝑛𝑜𝑑𝑒 is the potential of the anode [3].

As aforementioned, the electropositive nature of lithium contributes to the high voltages of

Li-ion batteries. For applications such as grid-scale storage, it is ideal to use batteries with the

highest voltage possible so that fewer cells are necessary [4].

Charge capacity (𝐶𝑐𝑒𝑙𝑙) is the amount of electric charge available in the battery, as shown

in equation 2,

𝐶𝑐𝑒𝑙𝑙 = (𝐶𝑐−1 + 𝐶𝑎

−1)−1, (2)

where 𝐶𝑐 and 𝐶𝑎 are the specific capacities of the cathode and anode, respectively [3]. The

amount of available charge depends on the amount of materials in the battery [4].

Energy density (𝐸𝑑) is the energy per total volume of the anode and cathode materials, as

given by equation 3,

𝐸𝑑 =𝐸𝑛𝑒𝑟𝑔𝑦

𝑉𝑜𝑙𝑢𝑚𝑒=

𝐶𝑐𝑒𝑙𝑙×𝑉𝑐𝑒𝑙𝑙×(𝑊𝑐+𝑊𝑎+𝑊𝑒)𝑊𝑐𝜌𝑐

+𝑊𝑎𝜌𝑎

+𝑊𝑒𝜌𝑒

, (3)

where 𝑊𝑐/𝑎/𝑒 and 𝜌𝑐/𝑎/𝑒 are the weight and density of the cathode/anode/electrolyte,

respectively [3].

Finally, the characteristic that this thesis will focus on is power density, the amount of

power per unit volume [4]. Since power is defined as the product of voltage and current and

the voltage of a battery is effectively constant, power density is determined by the amount of

current that can be drawn from a battery (For the purposes of this thesis, current will be

referred to as the flow of electrons).

4

For example, electric vehicles need batteries with high power density for rapid charge and

discharge [5]. As illustrated by Figure 1, the speed of the electrons through the load is limited

by the speed of their positive charge carrier counterparts through the battery’s electrolyte. The

system will maintain its charge neutrality, and therefore, the power that can be extracted from

a battery depends on how quickly and easily positive ions can travel through the electrolyte,

otherwise known as the ionic conductivity of the electrolyte.

1.2 Ion Transport in Batteries

Ions move through the electrolyte by hopping between interstitial sites [6]. Interstitial sites

are the normally unoccupied spaces or vacancies in a crystal structure that are typically the

most energetically stable. There are two varieties of interstitial sites that will be considered in

this thesis: tetrahedral (T-site) and octahedral (O-site). A tetrahedral site is a vacancy that is

formed by four atoms in the shape of a tetrahedron (Figure 2a), and similarly, an octahedral

site is a vacancy that is formed by six atoms in the shape of an octahedron (Figure 2b). This

thesis will show how the size and configuration of these interstitial sites determine the ionic

conductivity of each crystal structure.

Figure 2. Tetrahedral and an octahedral interstitial site. The tetrahedral site (a) has four anions around a cation,

and the octahedral site (b) has six anions around a cation. Anions in blue, cations in gold.

5

Conventional batteries contain liquid organic electrolytic solutions, and although they

possess desirable ionic conductivities in the order of 1 [mS cm-1], they are plagued by high

flammability [6]. Conversely, electrolytes made of inorganic solid materials are of interest

because they are electrochemically stable and not flammable [6]. The only solid-state battery

that is presently viable on an industry level is a sodium-sulfur battery than can only be used at

high temperatures (~250-300 [°C]) [7]. Clearly, solid-state electrolytes are hindered by generally

low ionic conductivities and require further research and development, both experimental and

computational, to become truly competitive with conventional liquid electrolytes.

That being said, several solid-state electrolytes have demonstrated remarkably high ionic

conduction and are thusly known as “superionic conductors.” Regarding Li-ion conductors,

oxide-based superionic conductors have shown ionic conductivities in the range of 10-3 to 1

[mS cm-1], and sulfide-based superionic conductors have shown ionic conductivities above 1

[mS cm-1] [6]. These numbers are comparable to those of liquid electrolytes (in the range of 1

[mS cm-1]) and bode well for the future commercialization of Li-ion solid-state electrolytes [6].

Lithium-ion technology will likely be the leader of the solid-state battery industry due to

the exceptional mobility of Li-ions in many solids [8]. However, growing demand and possible

scarcity of lithium has led to extreme price increases and narrowing profit margins for Li-ion

battery manufacturers [9]. To combat the potential resource and cost issues of lithium,

sodium-ion technologies are garnering interest because sodium is less expensive and much

more abundant than lithium. It is worth noting that lithium’s future in the battery industry is

highly debated. While many believe that lithium will become too scarce and, consequently, too

expensive, some studies suggest that there exist sufficient lithium reserves to maintain prices

at reasonable levels [10]. Regardless, it is imperative to research tangentially lithium-ion

technologies, which are clearly superior and promising in their performance abilities, as well

6

as other alternative materials like sodium that will be necessary for producing environmentally

sustainable and cost-effective batteries, especially if the era of lithium wanes.

1.3 Using Density Functional Theory to Investigate Ionic Conductivity

The aim of this thesis is to investigate the relationship between ionic conductivity and

crystal structure in solid-state electrolytes. This study of ionic conductivity is performed by

analyzing the activation energy for each crystal structure as structural energy is linked to ionic

conductivity through the relationship shown in equation 4,

σ = Ae−∆E

kbT , (4)

where 𝜎 is ionic conductivity, ∆𝐸 is activation energy, T is temperature, A is the pre-

exponential factor, and kb is the Boltzmann constant. Physically, the activation energy is the

energy needed for an ion to hop from one interstitial site to another (as shown in Figure 4),

and mathematically, ionic conductivity increases as the activation energy of a crystal structure

decreases (as shown in equation 1).

The activation energies of solid crystal structures can be determined via Density

Functional Theory (DFT). Based on the Schrödinger Equation, DFT is a quantum approach

for finding the ground-state energy of a crystal structure using its electron density [11]. For

the scope of this project, a deep understanding of how DFT operates is not necessary. Figure

3 shows a general overview for how DFT energy relaxations are performed.

7

Figure 3. High-level overview of the algorithm used for DFT energy relaxations. Note that |𝐹𝑖 − 𝐹𝑖−1| < 𝐹𝑐𝑢𝑡𝑜𝑓𝑓

means the forces of the previous iteration (𝑖 − 1) subtracted from the current iteration 𝑖 must be below a

specified cutoff value.

When combined with the Nudged Elastic Band (NEB) theory, DFT can be used to identify

superionic conductors. The NEB method involves calculating the ground state energy of an

ion at every point along its migration path through a solid crystal. For this thesis, NEB theory

was applied to the path of a cation hopping between interstitial sites in an anion lattice to

produce energy landscapes, examples of which are shown in the Li-ion migration plots from

Figure 4. From these energy landscapes, the activation energy for each structure can be

obtained (where ∆𝐸 is the largest change in energy for each plot), and thus, the ionic

conductivity of each structure relative to the others can be determined.

8

Figure 4. Crystal structure models and energy landscape plots from “Design principles for solid-state lithium

superionic conductors” by Wang et al. for BCC, FCC, and HCP crystal structures composed of a sulfur anion

lattice (yellow) and Li-ion charge carriers (green) [6].

In their 2015 paper “Design principles for solid-state lithium superionic conductors”,

Wang et al. used Density Functional Theory and Nudged Elastic Band theory to find a

correlation between crystal structure and ionic conductivity. Their work is built on the premise

that ionic conductivity is primarily linked to the shape of the anion sublattice of a material.

This means that DFT calculations do not need to be performed on an accurate model of an

actual existing material, which may be incredibly complex, but rather on a simplified model of

9

the structure that only contains an anion framework and mobile cations. Figure 5 shows how

Li10GeP2S12, an existing superionic conductor, is mapped to a simplified anion sublattice model.

Because DFT calculations require expensive high-performance computing power, simplifying

models is cost- and time-effective and limits the number of variables in a system.

Figure 5. Anion sublattice mapping. a) Existing superionic conductor Li10GeP2S12 (LGPS) [12]. b) LGPS with

Ge (blue) and P (green) removed. c) LGPS ultimately maps closely to a single bcc unit cell of sulfur anions

with one lithium ion in a tetrahedral site [13]. Sulfur anions in yellow, Li-ions in purple.

1.4 Project Statement & Scope

The goal of this thesis is to replicate the work of Wang et al. and to confirm the following

salient conclusions from their work: first, that body-centered cubic crystal structures are the

most ideal for ionic conduction due to their homogenous tetrahedral interstitial sites and,

second, that an anion sublattice is the most important physical characteristic contributing to

ionic conductivity and is therefore sufficient for modelling a crystal structure. The purpose of

verifying these powerful conclusions is to contribute to a more widespread use of DFT in

battery materials research and to subsequently accelerate the progress of battery materials

R&D.

10

Chapter Two – Methods

2.1 Selecting Crystal Structures

The first step for any structural energy investigation is to build the crystal structures in

question, which was completed in the Atomic Simulation Environment [13]. To replicate the

results of Wang et al., body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal

close packing (HCP) crystal structures were constructed for seven different volumes: 28.5, 34,

40, 46.6, 54, 62.1, and 70.8 [Å3] per S atom. Note that Wang et al. used volume per S atom

because their research focused on Li-ions moving through S anion sublattices. For this thesis,

the seven volumes were used for both sulfur anion sublattices as well as oxygen anion

sublattices (hence, volume per O atom). These crystal structures and volumes were applied to

four different chemistries: lithium ions in a sulfur lattice (Li-S), lithium ions in an oxygen lattice

(Li-O), sodium ions in a sulfur lattice (Na-S), and sodium ions in an oxygen lattice (Na-O). In

summary, four chemistries, three structures, and seven volumes results in 84 different cases to

be investigated.

It is also important to note that these structures are not necessarily based on materials that

exist in the real world. As described in Section 1.3 Using Density Functional Theory to

Investigate Ionic Conductivity, it is only necessary to perform DFT calculations on the anion

sublattice framework within a material to make inferences about that material’s ionic

conductivity. So, regardless of whether the crystal structures in this study are usable in batteries

themselves, the knowledge gained about how particular frameworks affect the ionic

conductivity of actual materials will be useful in identifying and designing superionic

conductors.

11

2.2 Interstitial Sites and Ion Hopping Paths

The second step in the DFT energy calculation process is to perform ground-state energy

relaxation calculations when the cations are in the anion lattice interstitial sites. The interstitial

sites should be stable locations where the energy of the structure is the lowest locally. Since

the ions move through solid electrolytes by hopping between interstitial sites, each interstitial

site can be thought of as the beginning and endpoints, or the initial and final states, of any

migration path. The NEB calculation results are used to plot the change in energy as an ion

moves from one interstitial site to the next, so the ground-state energies of the initial and final

states are necessary precursors to interpolating the migration path between interstitial sites.

For the BCC structures, there is only one possible path through the unit cell: T1-T2. The

T-T movement in BCC structures is illustrated in Figure 6. This means the ion hops directly

from tetrahedral site T1 to tetrahedral site T2. For each BCC structure, only one energy

landscape will be generated because there is only one possible path for movement.

Figure 6. Li-ion migration between tetrahedral sites T1 and T2 in a primitive BCC sulfur cell.

For the FCC structures, there is only one possible path through the unit cell: T1-O1-T2.

The T-O-T movement in FCC structures is illustrated in Figure 7. This means the ion must

hop from the T1 site to the O1 site before returning to the T2 site. Note that a direct path

from the T1 site to the T2 site was not considered as it was assumed that the energy barrier

would be too prohibitive for ion diffusion [6]. When moving from the T1 to the O1 position,

12

only one bond will be broken, but when moving directly from T1 to T2, two bonds must be

broken, resulting in a higher activation energy. For each FCC structure, only one energy

landscape will be generated because there is only one possible path for movement.

Figure 7. Li-ion migration from tetrahedral site T1 to octahedral site O1 and finally to tetrahedral site T2 in a

primitive FCC sulfur cell.

For HCP structures, there are three possible paths through the unit cell: T1-O1-T2, T1-

T3, and O1-O2. All migration paths within an HCP structure are illustrated in Figure 8. As in

BCC and FCC structures, ions can move in a T-O-T path or a T-T path, but they can also

move from one octahedral site to another in an O-O path. For each HCP structure, three

energy landscapes will be generated because there are three possible paths for movement.

Figure 8. Li-ion migration in a primitive HCP sulfur cell. There are three paths for ionic interstitial hopping:

tetrahedral to tetrahedral (T1-T3), tetrahedral to octahedral to tetrahedral (T1-O1-T2), and octahedral to

octahedral (O1-O2).

13

To summarize, for the four different chemistries, seven volumes were considered. For

those seven volumes, three different crystal structures were investigated. And for those three

structures, the energy landscape for each possible path (T-T, O-O, T-O-T) were calculated.

2.3 DFT Workflow

The workflow for performing DFT calculations begins with using the Atomic Simulation

Environment to generate the crystal structures. Because the DFT calculations were performed

using the Vienna Ab Initio Simulation Package (VASP), the crystal structures were written to

a VASP file called a POSCAR.

The VASP pseudopotentials that contain all the information about each element in the

crystal structure are saved in the POTCAR file. Depending on the case being calculated, this

file contains information on either Li and S, Li and O, Na and S, or Na and O. For this thesis,

these POTCARs have been altered to reflect the ions for each element rather than the atom.

For example, the POTCAR would normally read that sulfur has 6 valence electrons and Li has

1, but for these calculations, it has been modified to show that sulfur has 8 valence electrons

(to make S2-) and Li has 0 (to make Li+).

All the desired calculation settings and parameters are written to a VASP file called the

INCAR; the specific parameters used can be found in the Appendix. It is highly relevant to

note that in order to perform calculations on an anion lattice, it is necessary to use the INCAR

file to add a uniform background charge to the cell. VASP requires a neutral system, which is

not useful for modelling solid-state electrolytes because the application of an electrolyte in a

battery in and of itself implies that the ions must be charged. To achieve an accurate charge

that is still compatible with VASP’s operation requirements, a uniform background charge is

assumed. For example, in the Li-S BCC case, the realistic, charged model would have 16

valence electrons (8 per S2-, 0 per Li+), and the neutral model would have 13 valence electrons

14

(6 per S, 1 per Li). In order to balance the system, the NELECT tag is set to 13 in the INCAR

file, so VASP will subtract 13 from 16 to add 3 positive background charges to the calculation.

Figure 9 provides a visual representation for how the uniform background charge is

implemented.

Figure 9. Illustration of how VASP incorporates a uniform background charge.

To put it broadly, DFT relaxation calculations rely on an iterative process of using the

electronic density of a crystal structure to calculate the energy of the configuration, checking

if the forces in the system are low enough to be considered converged, and then either exiting

or repeating the process (refer to Figure 3). During this study, it was discovered that in order

for VASP to make correct initial guesses for the atom positions when the uniform background

charge was in use (henceforth to be referred to as the “charged system”), VASP was

inexplicably unable to generate an initial guess for the atom positions. Thus, it was determined

that a WAVECAR, a VASP file containing information on the atom positions, must be

supplied as an input to the calculation. The WAVECAR was generated by first performing a

ground-state energy calculation on a neutral system (no altered POTCARs and no uniform

15

background charge) to determine initial atom positions, and then, this WAVECAR was used

in all charged system DFT calculations.

2.4 Nudged Elastic Band Calculations

Once the ground-state energies for the initial and final states had been calculated, NEB

theory was applied to find the energy landscapes for an ion migration path. First, a physical

path between the initial and final states was approximated, as shown in Figure 10.

Figure 10. Example of how an Li-ion migration path is interpolated for an NEB calculation.

Second, as with the ground-state calculations, the WAVECAR files from the initial and

final state calculations are copied into the NEB folders. This is necessary so that VASP can

set reasonable starting positions for the NEB.

Third, the NEB calculations are run. The calculations finally finished when the forces were

lower than the EDIFFG value, a tag in the INCAR file that denotes the exit condition for the

ionic relaxation loop; it was frequently necessary to rerun the calculations many times for the

NEBs to fully converge.

16

Chapter Three – Results & Discussion

3.1 Lithium Cations in a Sulfide Framework (Li-S)

3.1.1 Li-S Body-Centered Cubic

Figure 11 shows the energy landscapes of a lithium ion in a BCC sulfur anion sublattice.

The general parabolic shape of these curves is intuitive (and consistent with the shape of the

energy landscapes produced by Wang et al., shown in Figure 12): the energies are the lowest at

the tetrahedral sites, and the energy is the highest at the midpoint between the tetrahedral sites

where the interatomic forces are the strongest (refer to Figure 6).

Figure 11. Energy landscapes for a Li-ion migrating through a BCC S anion sublattice via a T-T path.

Figure 12. Findings from Wang et al. of the energy landscapes for a Li-ion migrating through a BCC S anion

sublattice via a T-T path [6].

17

3.1.2 Li-S Face Centered Cubic

Figure 13 shows the energy landscapes of a lithium ion in an FCC sulfur anion sublattice.

The shapes of the energy landscapes are reasonable considering the path the ion must take in

an FCC structure (refer to Figure 7). The ion begins at a tetrahedral site and overcomes a small

energy barrier to move to an octahedral site, where it settles. Then, from the octahedral site,

the ion must overcome another energy barrier to move to its final tetrahedral site.

Figure 13. Energy landscapes for a Li-ion migrating through an FCC S anion sublattice via a T-O-T path.

While the shapes of the curves in Figure 13 seem reasonable, there is a slight inconsistency

between these results and those produced by Wang et al., seen in Figure 14. Wang et al.’s results

show that all energy landscapes for calculations that used a unit cell volume of ≥46.6 [Å3] per

S should be concave-down with the highest energy barrier at the central octahedral site. The

18

curves in Figure 13, which all correspond to structures with volumes above 46.6 [Å3] per

S, seem to exhibit the energy landscapes for less voluminous cells.

Figure 14. Findings from Wang et al. of the energy landscapes for a Li-ion migrating through an FCC S anion

sublattice via a T-O-T path [6].

3.1.3 Li-S Hexagonal Close Packing

Figure 15 shows the energy landscapes of a lithium ion in an HCP sulfur anion sublattice

via the T-T migration path. The ion begins at a tetrahedral site and overcomes an energy

barrier to move to its final tetrahedral site. It was expected that the initial and final state

energies would be the same with a maximum energy barrier at the midpoint between the

tetrahedral sites, as in the results from Wang et al. shown in Figure 16.

Figure 15. Energy landscapes for a Li-ion migrating through an HCP S anion sublattice via a T-T path.

19

Figure 16. Findings from Wang et al. of the energy landscapes for a Li-ion migrating through an HCP S anion

sublattice via a T-T path [6].


via the O-O migration path. The ion begins at an octahedral site and overcomes an energy

barrier to move to its final octahedral site. The energy landscapes reflect the expected

energetics and are similar to the results from Wang et al. shown in Figure 18.

Figure 17. Energy landscapes for a Li-ion migrating through an HCP S anion sublattice via an O-O path.

20


sublattice via an O-O path [6].


via the T-O-T migration path. Like the FCC structures, the Li-ion begins at a tetrahedral site

and overcomes an energy barrier to move to an octahedral site, where it settles, and from the

octahedral site, the ion must overcome another energy barrier to move to its final tetrahedral

site.

Figure 19. Energy landscapes for a Li-ion migrating through an HCP S anion sublattice via a T-O-T path.

21

Figure 20 shows the energy landscapes produced by Wang et al. for a Li-ion migrating

through an HCP sulfur anion sublattice following at T-O-T path.


sublattice via a T-O-T path [6].

3.1.4 Activation Energy for Li-ion Migration in S Anion Framework vs. Volume

Identifying trends between energy and cell volume is also important as there is, or should

be, a range of volumes for which the energy of a crystal structure is optimized. Typically, it is

expected that larger cell volumes will have the lowest energy barriers. When the atoms in a

structure are relatively far apart, their interatomic forces are weaker, and thus, less energy is

required to overcome said forces.

Figure 21 shows the correlation between activation energy and unit cell volume for Li-

S structures. These curves are consistent with the hypothesis that larger volumes have lower

activation energies due to lower interatomic forces; this is consistent for all crystal structures

(BCC, FCC, & HCP) and for all energy pathways (T-T, O-O, & T-O-T). In addition, FCC

structures are also shown to have higher activation energies than BCC structures across all

volumes. FCC unit cells contain more atoms than BCC cells, which increases that amount of

interatomic force that a Li-ion would experience while migrating through the structure.

22

Therefore, it is understandable that FCC structures would have higher activation energies than

BCC structures overall.

Interestingly, the TT path in HCP structure has comparable energy-volume relationships

to the BCC structures. This shows that direct movement between tetrahedral sites of

equivalent energy is ideal for ion diffusion, regardless of whether the anion sublattice is a BCC

or HCP structure.

Figure 21. Energy barriers versus cell volume per S anion for various Li-S crystal structures. The dotted lines

show the expected trends.

For comparison, Figure 22 shows the plots for energy barriers (activation energy) versus

cell volume for Li-S structures generated by Wang et al. Their plots show the same three

phenomena: first, FCC structures have higher activation energies for all volumes; second,

activation energies are generally lower for higher volumes; and third, ion migration paths that

move directly between interstitial sites of equivalent energies (T-T or O-O) typically have the

lowest energy barriers.

23

Figure 22. Findings from previous work of the energy barriers versus cell volume per S anion for various Li-S

crystal structures [6].

3.2 Sodium Cations in a Sulfide Framework (Na-S)

3.2.1 Na-S Body-Centered Cubic

Like the Li-S BCC energy landscapes, the energy landscapes for sodium ions migrating

through a sulfur anion sublattice, seen in Figure 23, are parabolic-shaped. The similarities

between the Li-S and Na-S BCC energy landscapes serve as evidence that elements within the

same group on the periodic table (and thus with the same number of valence electrons) exhibit

similar ion diffusion characteristics. Though, it should also be noted that the activation

energies of the Li-S BCC cases were lower than those in the Na-S BCC cases; this is likely due

to the relative atomic masses (6.94 [u] for Li and 22.99 [u] for Na) and radii (182 [pm] for Li

24

and 227 [pm] for Na) of lithium and sodium. Lithium, being lighter and less voluminous,

requires less energy to move between tetrahedral sites in a sulfur framework.

Figure 23. Energy landscapes for a Na-ion migrating through a BCC S anion sublattice via a T-T path.

3.2.2 Na-S Face-Centered Cubic

Figure 24 shows the energy pathway of a sodium ion in an FCC sulfur anion sublattice.

Figure 24. Energy landscapes for a Na-ion migrating through an FCC S anion sublattice via a T-O-T path.

The energy landscapes of sodium ions migrating through a sulfur anion sublattice, seen in

Figure 24, have a similar shape to the Li-S FCC landscapes in Figure 13. The energies are at

local minima at the tetrahedral and octahedral sites and reach local maxima when moving from

T1 to O1 or O2 to T2. As with the BCC cases, the similarities between the Li-S and Na-S FCC

25

cases suggest that positive charge carriers from the same period table group diffuse through

crystal structures in the same way.

3.2.3 Na-S Hexagonal Close Packing

Figure 25 shows the energy landscapes of a sodium ion migrating through an HCP sulfur

anion sublattice via the T-T migration path. The ion begins at a tetrahedral site and overcomes

an energy barrier to move to its final tetrahedral site, so it was expected that the initial and

final state energies would be the same with a maximum energy barrier at the midpoint between

the tetrahedral sites. This expectation was confirmed in the energy landscapes for the 54 and

62.1[Å3] per S cases.

Figure 25. Energy landscapes for a Na-ion migrating through an HCP S anion sublattice via a T-T path.

Figure 26 shows the energy landscapes of a sodium ion migrating through an HCP sulfur

anion sublattice via the O-O migration path. Like the T-T pathway, it was expected that the

initial and final state energies would be the same with a maximum energy barrier at the

midpoint between the octahedral sites, and also like the T-T results in Figure 25, the 54 and

62.1 [Å3] per S cases exhibit the expected behavior.

26

Figure 26. Energy landscapes for a Na-ion migrating through an HCP S anion sublattice via an O-O path.

Figure 27 shows the energy pathway of a sodium ion migrating through an HCP sulfur

anion sublattice via the T-O-T migration path.

Figure 27. Energy landscapes for a Na-ion migrating through an HCP S anion sublattice via a T-O-T path.

3.2.4 Activation Energy for Na-ion Migration in S Anion Framework vs. Volume

Figure 28 shows the correlation between activation energy and unit cell volume for the

Na-S cases. The results of Figure 28 remain consistent with previous conclusions: BCC

structures have lower activation energy barriers than FCC structures, and activation energies

27

decrease as cell volume increases. For comparison, Figure 29 shows the plots for activation

energy versus cell volume for Na-S structures generated by Wang et al.

Figure 28. Energy barriers versus cell volume per S anion for various Na-S crystal structures.

Figure 29. Findings from Wang et al. of the energy barriers versus cell volume per S anion for various Na-S crystal

structures [6].

3.3 Lithium Cations in an Oxide Framework (Li-O)

3.3.1 Li-O Body-Centered Cubic

Figure 30 shows the energy landscapes of a lithium ion through a BCC oxygen anion

sublattice. While the curves may seem to differ greatly from the clean parabolic shapes from

the BCC sulfide cases, the differences are in the range of 10-3 [eV], which is most likely within

the accuracy of the calculation. In general, the energy landscapes begin and end around 0 [eV].

28

For the large volumes (≥46.6 [Å3] per O), disregarding the outlying data point in the 70.8 [Å3]

per O curve, the energy landscapes are relatively flat, suggesting that Li-O BCC structures

might make for a promising superionic conductor.

Figure 30. Energy landscapes for a Li-ion migrating through a BCC O anion sublattice via a T-T path.

3.3.2 Li-O Face-Centered Cubic

Figure 31 shows the energy landscapes of a lithium ion in an FCC oxygen anion sublattice.

Figure 31. Energy landscapes for a Li-ion migrating through an FCC O anion sublattice via a T-O-T path.

3.3.3 Li-O Hexagonal Close Packing

Figure 32 shows the energy landscapes of a lithium ion in an HCP oxygen anion sublattice

via the T-T migration path.

29

Figure 32. Energy landscapes for a Li-ion migrating through an HCP O anion sublattice via a T-T path.

Figure 33 and Figure 34 show the energy pathways of a lithium ion in an HCP oxygen

anion sublattice via the O-O and T-O-T migration paths, respectively.

Figure 33. Energy landscapes for a Li-ion migrating through an HCP O anion sublattice via an O-O path.

30

Figure 34. Energy landscapes for a Li-ion migrating through an HCP O anion sublattice via a T-O-T path.

3.3.4 Activation Energy for Li-ion Migration in O Anion Framework vs. Volume

Figure 35 shows the correlation between activation energy and unit cell volume for the Li-

O cases. The Li-O FCC and HCP energy vs. volume results are excluded given that the energy

landscapes on which they are based are unrealistic. Regardless, the relationship between

activation energy and volume is realistic for the Li-O BCC cases and has a similar curvature to

Wang et al.’s findings, shown in Figure 36.

Figure 35. Energy barriers versus cell volume per O anion for various Li-O crystal structures.

31

Figure 36. Findings from Wang et al. of the energy barriers versus cell volume per O anion for various Li-O

crystal structures [6].

3.4 Sodium Cations in an Oxide Framework (Na-O)

3.4.1 Na-O Body-Centered Cubic

Figure 37 shows the energy pathways of a sodium ion in a BCC oxygen anion sublattice.

These curves are intuitive in that they all have a general parabolic shape, begin and end at 0

[eV], and have increasing maxima as cell volume decreases. However, it is surprising that the

intermediate position was found to have a lower energy than the endpoint tetrahedral sites for

the 54, 62.1, and 70.8 [Å3] per O cases. Though, as mentioned for the Li-O cases, the order of

the energies suggests that this deviation is likely within the accuracy of the calculations. The

similarity of the Na-O BCC landscapes to both the Li-S and Na-S BCC landscapes support

32

the idea that anion lattices consisting of elements from the same periodic groups have similar

structural energies.

Figure 37. Energy landscapes for a Na-ion migrating through a BCC O anion sublattice via a T-T path.

3.4.2 Na-O Face-Centered Cubic

Figure 38 shows the energy landscapes of a sodium ion in an FCC oxygen anion sublattice.

Figure 38. Energy landscapes for a Na-ion migrating through an FCC O anion sublattice via a T-O-T path.

Figure 39, Figure 40, and Figure 41show the energy pathways of a sodium ion in an HCP

oxygen anion sublattice via the T-T, O-O, and T-O-T pathways, respectively.

33

Figure 39. Energy landscapes for a Na-ion migrating through an HCP O anion sublattice via a T-T path.

Figure 40. Energy landscapes for a Na-ion migrating through an HCP O anion sublattice via an O-O path.

Figure 41. Energy landscapes for a Na-ion migrating through an HCP O anion sublattice via a T-O-T path.

34

3.4.4 Activation Energy for Na-ion Migration in O Anion Framework vs. Volume

Figure 42 shows the correlation between activation energy and unit cell volume for the

Na-O cases. The Na-O FCC and HCP curves were excluded as those energy landscapes are

unrealistic. The Na-O BCC energy vs. volume curve is as expected with activation energy

decreasing as volume increases.

Figure 42. Energy barriers versus cell volume per O anion for various Na-O crystal structures.

3.5 Limitations of Current Results

While many of the results in this thesis follow expected trends, there are a few cases that

have produced surprising and unrealistic data. For those perplexing cases, the following

inferences have been made regarding their causes and possible remedies.

First, for all the energy landscapes, there is a notable difference in the range of the energies

seen on the y-axis from what is expected. The energies in this thesis are approximately an

order of 10 times smaller than those produced by Wang et al. This is likely the result of a

difference in DFT calculation techniques. In particular, the results for this thesis were

generated by performing calculations on singular primitive unit cells, while Wang et al. used

3x3x3 supercells, which are the same unit cells multiplied three times in each direction. Using

only a single unit cell may have distorted the forces, and thus the energies of the crystal

35

structures, whereas a 3x3x3 supercell is better representative of the interatomic forces that

would be present throughout a real material.

Second, the FCC and HCP oxide cases showed energy landscapes that were

counterintuitive. The results presented did not take into account the electron spin of the

oxygen valence electrons, but it is thought that this many have an effect on the energies. For

example, the occupation of the orbitals for oxygen atoms determines the bandgap energies. If

these bandgap energies are too large (which was the case for the generated results), then the

electron density that VASP assumes will likely be incorrect, ultimately resulting in an incorrect

final energy calculation.

Third, the smearing function, which determines the electron distribution used by VASP,

shall be changed. As shown in the Appendix, an ISMEAR value was not set in the INCARs.

It is believed that changing the smearing function to the Tetrahedron method with Blöchl

corrections (ISMEAR = -5) will improve the results as this is the preferred setting for

insulators.

Chapter Four – Summary and Conclusions

In conclusion, density functional theory coupled with nudged elastic band theory was used

to study the structural energies of solid-state electrolytes. Density functional theory

calculations were performed for body-centered cubic, face-centered cubic, and hexagonal

close packing structures for four chemistries (Li-S, Na-S, Li-O, Li-O) over a range of seven

volumes. Combing DFT with nudge elastic band theory produced energy landscapes, which

showed the energy necessary for a cation to hop between interstitial sites for each of the

chemistry/structure/volume cases. The energy landscapes revealed three major conclusions:

36

First, the activation energies for ions migrating through BCC structures were consistently

lower than ions migrating through FCC and HCP structures across all chemistries and

volumes, though the HCP structures for the T-T and O-O cases were quite comparable. This

finding shows that BCC anion sublattices are ideal for Li-ion transport due to their

homogenous tetrahedral sites. Therefore, materials with crystallographically equivalent

substructures should be favored by designers, but if this is not achievable, a heterogenous

structure in which a mobile cation has the possibility of hopping via T-T and/or O-O paths

would be the next best option.

Second, energy barriers tend to decrease as unit cell volume increases. This suggests that,

in general, a larger unit cell volume is desirable as it lowers the energy required for a migrating

cation to overcome the interatomic forces. Materials designers will need to find a balance

between optimizing unit cell volume for ion diffusion purposes and maintaining a practical

battery size.

Third, the energy landscapes for all four chemistries were similar. This suggests that

elements in the same groups on the period table (and thus with the same number of valence

electrons) have either similar transport mechanisms (Li and Na) or structural properties (S and

O). However, this study only accounted for cations in the alkali metal group and anions in the

chalcogen group, so it is possible that crystal structures with components from other groups

could have similar structural energies. Regardless, the hypothesis that Li- and Na-ions could

diffuse through S and O lattices in similar ways was proved to be true. More investigation is

necessary to definitively say whether transport properties are affected by the valency of the

structural components.

The breadth of this project could be expanded in a variety of ways. Investigating more

chemistries is an obvious route; two elements of particular interest are Mg2+ and Se2-. Mg2+ is

37

interesting because having a charge carrier that supplies 2 valence electrons would double the

charge that could be extracted from a typical Li-ion battery. However, designing a Mg-ion

electrolyte is especially challenging due to the size of the Mg-ion. In addition, Se2- is also worth

investigating because according to a study by Canepa et al., Se2- anion sublattices have even

lower energy barriers than oxides or sulfides and may be especially applicable for Mg-ion

diffusion [14].

The purpose of this thesis was to be a proof-of-concept for using ab initio simulations for

solid-state electrolyte design. Although batteries are one of the most necessary and important

technologies for energy sustainability, the rate at which they are developed and improved lags

far behind what is currently needed. For batteries, adhering to Moore’s Law, that is to say

doubling performance metrics every 18 months, is likely an unattainable goal, but significant

accelerations in progress can be made by implementing high performance computing more

frequently in battery materials research. This thesis shows that density functional theory

calculations can provide reasonable and repeatable results, though there can be inconsistencies

due to calculation settings, inaccurate modeling, etc. Therefore, it is recommended that future

solid-state electrolyte research utilize DFT as a precursor to all experimental research, serving

as a filter that determines where physical resources and time shall be allocated in the

experimental phase.

38

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batteries/. [Accessed 1 February 2019].

41

Appendix

DFT Python script for an uncharged Li-S BCC system with Li in the T1 position

from ase.io import write from ase.build import bulk from ase import Atom from ase.calculators.vasp import Vasp import os os.system('cp ../volume .') f = open('volume','r') unit_cell = f.read() no_S_atoms = 2 #number of anions in the structure cell_volume = float(unit_cell)*no_S_atoms side = cell_volume**(1.0/3.0) #For T2, x=0, y=side/2, and z = side/4 x=0 y=side/4 z=side/2 anion = bulk('S','bcc',a=side,covera=1,cubic=1) # ‘S’ can be changed to a different element cation = Atom('Li',[x,y,z]) # ‘Li’ can be changed to a different element atoms = anion + cation #building the structure write('POSCAR',atoms,format='vasp') #writing the structure to a VASP file #setting calculation parameters calc = Vasp( xc='PBE', encut=500, ibrion=2, npar=4, kpts=(6,6,6), ismear=-5, ispin=2, algo='Conjugate', ediffg=-0.2, nsw=0) calc.initialize(atoms) atoms.set_calculator(calc) calc.write_incar(atoms) #writes INCAR file calc.write_kpoints() #writes KPOINTS file os.system('sbatch submit.sh')

42

DFT Python script for an uncharged Li-S FCC system with Li in the T1 position

from ase.io import write from ase.build import bulk from ase import Atom from ase.calculators.vasp import Vasp import os os.system('cp ../volume .') f = open('volume','r') unit_cell = f.read() no_S_atoms = 4 cell_volume = float(unit_cell)*no_S_atoms side = cell_volume**(1.0/3.0) # For O1, x=y=z=side/2 # For T2, x= y=z=side/4 x=side/4 y=side/4 z=3*side/4 anion = bulk('S','fcc',a=side,covera=1,cubic=1) cation = Atom('Li',[x,y,z]) atoms = anion + cation write('POSCAR',atoms,format='vasp') calc = Vasp( xc='PBE', encut=500, ibrion=2, npar=4, kpts=(6,6,6), ismear=-5, ispin=2, algo='Conjugate', ediffg=-0.2, nsw=0) calc.initialize(atoms) atoms.set_calculator(calc) calc.write_incar(atoms) calc.write_kpoints() os.system('sbatch submit.sh')

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DFT Python script for an uncharged Li-S HCP system with Li in the T1 position

from ase.io import write from ase.build import hcp0001 from ase import Atoms from ase.calculators.vasp import Vasp import os import math from ase.io.vasp import write_vasp os.system('cp ../volume .') f = open('volume','r') A = (float(f.read())*(2.0)**(1.0/2.0))**(1.0/3.0) S = hcp0001('S',size=(2,2,3),a=A,orthogonal=True) Li = Atoms('Li') cell1 = S.get_cell() cell1[2][2] = S.get_positions()[-1][2] S.set_cell(cell1) del S[[8,9,10,11]] # For T2, Li.set_positions([(S.get_positions()[1]+S.get_positions()[2]+S.get_positions()[5]+ # S.get_positions()[3])/4]) # For T3, Li.set_positions([(S.get_positions()[10]+S.get_positions()[4]+S.get_positions()[5]+ # S.get_positions()[6])/4]) # For O1, Li.set_positions([(S.get_positions()[0]+S.get_positions()[5])/2]) # For O2, Li.set_positions([(S.get_positions()[8]+S.get_positions()[5])/2]) Li.set_positions([(S.get_positions()[2]+S.get_positions()[4]+S.get_positions()[5]+S.get_positions()[6])/4]) atoms = S + Li k1 = math.ceil(20/(cell1[0][0])) k2 = math.ceil(20/(cell1[1][1])) k3 = math.ceil(20/(cell1[2][2])) write('POSCAR',atoms,format='vasp') calc = Vasp( xc='PBE', encut=500, ibrion=2, npar=4, kpts=(k1,k2,k3), ismear=-5, ispin=2, algo='Conjugate',

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ediffg=-0.2, nsw=0) calc.initialize(atoms) atoms.set_calculator(calc) calc.write_incar(atoms) calc.write_kpoints() write_vasp('POSCAR', calc.atoms_sorted, symbol_count=calc.symbol_count) calc.write_sort_file() os.system('sbatch submit.sh')

INCAR parameters for DFT calculations on a charged BCC system

INCAR created by Atomic Simulation Environment ENCUT = 500.000000 NELECT = 13.000000 SIGMA = 0.100000 EDIFFG = -2.00e-01 GGA = PE NPAR = 4 IBRION = 2 NBANDS = 24 NSW = 500

INCAR parameters for DFT calculations on a charged FCC system

INCAR created by Atomic Simulation Environment ENCUT = 500.000000 NELECT = 25.000000 SIGMA = 0.100000 EDIFFG = -2.00e-01 GGA = PE NPAR = 4 IBRION = 2 NBANDS = 32 NSW = 500

INCAR parameters for DFT calculations on a charged HCP system

INCAR created by Atomic Simulation Environment ENCUT = 500.000000 NELECT = 49.000000

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SIGMA = 0.100000 EDIFFG = -2.00e-01 GGA = PE ALGO = Conjugate NPAR = 4 NELM = 200 IBRION = 2 NBANDS = 64 NSW = 500

Low Precision NEB calculation for a charged BCC system import os script = '''#!/usr/bin/env python #SBATCH -p batch #SBATCH -o myMPI.o%j #SBATCH -N 1 -n 27 #SBATCH -t 48:00:00 #SBATCH --mail-type=END #SBATCH [email protected] # Number of nodes = number of images not including the initial and final states import os from ase.io import read from ase.calculators.vasp import Vasp import shutil # Change according to system os.environ['VASP_COMMAND']='mpirun vasp_std' calc = Vasp(xc='PBE', ismear=-5, ispin=2, images=9, #Enter the number of images npar=4, # Change according to the system kpts=(6,6,6), encut=400, ediffg=-0.1, nsw=10000, nelect=13, # 25 for FCC; 49 for HCP nbands=24, # 32 for FCC, omit for HCP ibrion=3, spring=-5, potim=0.0, prec='Normal',

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ichain=0, iopt=3, algo='Conjugate', lwave=True, lcharg=False, lreal='False', lclimb=False) atoms = read('./01/POSCAR') atoms.set_calculator(calc) atoms.get_potential_energy() ''' wd = os.getcwd() with open(wd+'/filename-low-prec', 'w') as f: f.write(script) os.system('sbatch filename-low-prec') os.chdir(wd)

High Precision NEB calculation for a charged BCC system import os script = '''#!/usr/bin/env python # Change according to system # Number of nodes = number of images not including the initial and final states #SBATCH -p batch #SBATCH -o myMPI.o%j #SBATCH -N 1 -n 27 #SBATCH -t 48:00:00 #SBATCH --mail-type=END #SBATCH [email protected] from ase.io import read from ase.calculators.vasp import Vasp import shutil import os, glob cwd = os.getcwd() imagedirs = glob.glob('0?') imagedirs.sort() for imagedir in imagedirs[1:-1]: os.chdir(imagedir) os.system('cat POSCAR > POSCAR.old') os.system('cat OUTCAR > OUTCAR.old')

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os.system('cat CONTCAR > POSCAR') os.chdir(cwd) # Change accordingly os.environ['VASP_COMMAND']='mpirun vasp_std' calc = Vasp(xc='PBE', nsw=10000, ibrion=3, spring=-5, sigma=0.1, potim=0.0, prec='Normal', ichain=0, iopt=3, timestep=0.05, maxmove=0.05, algo='Fast', images=9, npar=4, lwave=False, lcharg=False, lreal='False', nelect=13, # 25 for FCC, 49 for HCP nbands=24, # 32 for FCC kpts=(6,6,6), encut=540, ediffg=-0.05, lclimb=True) atoms = read('01/POSCAR') atoms.set_calculator(calc) atoms.get_potential_energy() ''' wd = '.' with open(wd+'/filename-high-prec', 'w') as f: f.write(script) os.chdir(wd)

AN IN SOLID-STATE ELECTROLYTES

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